Sorry for the vague subject title, but I couldn't think of anything halfway between 'so vague there may as well be no title' and 'entire post is in the subject line'
Also, I couldn't find anything online to help me format this in LaTeX, so I'll just have to use plaintext.
Anyway, I learned about the summation function in school recently, and it got me thinking. I wondered if an expression of the form
Where c is a constant and n and x are variables could be rewritten as a function of n.
To test this, I asked wolfram alpha to sum from x=1 to x=n 2x, and it returned x^2 + x. I tried this with several other functions, including sin(x) and e^x, and it worked for them too. I did notice that there were some functions that didn't work with it, such as tan(x) and x^x. I also noticed that whenever f(x) was polynomial, f(n) was a polynomial with 1 extra power (eg, linear became quadratic, quadratic became cubic, x^20 became a very big function involving x^21).
What I was wondering was, how is this done? The only way I could think of to do this would be by some sort of regression using a bunch of different values for n, or in the case of the polynomials just taking however many points are required to define the curve.